Number

Worked examples (11)extended set

• 1Selecting numbers by type from a list
• 2Rational or irrational, with justification
• 3Converting a recurring decimal to a fraction
• 4Recurring decimal with a non-recurring part
• 5Smallest multiplier to make a perfect square
• 6Smallest multiplier to make a perfect cube
• 7Matching numbers to points on a number line
• 8Rational and irrational sides of tilted squares
• 9Cubes that rearrange into a square layer
• 10Natural, prime and irrational numbers in a mixed list
• 11Writing numbers in figures and in words

Worked examples (9)extended set

• 1Reading set notation
• 2Two-set Venn diagram problem
• 3Three-set Venn diagram
• 4Counting subsets
• 5The empty set and complements of sets defined by a rule
• 6Three sets of integers: complements and a shaded region
• 7Sets of numbers shown on number lines
• 8Completing a three-set Venn diagram of number types
• 9Sets of points on straight lines

Worked examples (13)extended set

• 1Writing a number as a product of its prime factors
• 2HCF and LCM from prime factorisation
• 3LCM in context — tiling a square
• 4LCM with a common remainder
• 5Using HCF × LCM = product of the numbers
• 6Finding unknown indices from HCF and LCM
• 7LCM of three numbers
• 8Completing a factor tree, then making a multiple
• 9HCF and LCM from a Venn diagram of prime factors
• 10Gear wheels and the lowest common multiple
• 11Choosing numbers from a list with a given product
• 12HCF and LCM of algebraic terms
• 13Factor pairs and positive integer solutions

Worked examples (11)extended set

• 1Sharing an amount in a given ratio
• 2Working backwards from a difference
• 3Combining two ratios
• 4Direct proportion
• 5Inverse proportion
• 6Proportional to a square
• 7Scale bar to map ratio, then a real area
• 8Mixing two drinks in different ratios
• 9Dividing a rectangle's area in a given ratio
• 10Simplifying a price ratio, then a fraction and a ratio together
• 11Areas on a scale plan

Worked examples (12)extended set

• 1Fraction, decimal and percentage of a shaded shape
• 2Dividing mixed numbers without a calculator
• 3Which fractions give terminating decimals?
• 4Adding two recurring decimals
• 5Percentages over 100% from a bar model
• 6Profit percentage after a unit conversion
• 7Percentage points against percent
• 8Working backwards and forwards from a sale label
• 9Checking decimal calculations without a calculator
• 10A bill with a percentage service charge
• 11Price before a cut, instalments and percentage extra
• 12Savings that grow by the same percentage each month

Worked examples (12)extended set

• 1Pay from a clock card
• 2Salary plus commission, worked backwards
• 3Simple interest: finding the time and the principal
• 4Payment plan, cash price and profit
• 5Payslip with pension and income tax
• 6Reading two savings accounts from a graph
• 7Depreciation at two different rates
• 8Paying off a loan with yearly repayments
• 9Overtime at a percentage above the basic rate
• 10Simple interest: total value and time to reach a target
• 11Finding a compound rate from two balances, then projecting forward
• 12Doubling time and the rate of compound interest

Worked examples (10)extended set

• 1Rounding to decimal places and significant figures
• 2Estimating a compound expression
• 3Estimating with a root and a power
• 4Deciding whether an estimate is too big or too small
• 5Choosing a sensible degree of accuracy
• 6Estimating a mean from rounded data
• 7Estimating the area of a field from its plan
• 8Tiling a wall: rounding a count in context
• 9Estimating around a running track
• 10Estimating a square root of a quotient

Worked examples (10)extended set

• 1Reading a timetable across midnight
• 2Spotting a decimal-hours mistake
• 3A train passing through a tunnel
• 4A flight across time zones
• 5Laps, speed and pace on a running track
• 6Two journeys on one distance–time graph
• 7Required speed on the final leg of a route
• 8Relative speed on a moving walkway
• 9An uphill and downhill ride with an unknown speed
• 10A river boat against and with the current

Worked examples (11)extended set

• 1Converting to and from standard form
• 2Multiplying and dividing in standard form
• 3Adding numbers in standard form
• 4Raising a standard-form number to a power
• 5Combining all four operations in standard form
• 6Standard form in context
• 7Real size from a microscope image
• 8Light travel time between Earth and Mars
• 9How many transistors fit on a chip?
• 10Roots of numbers in standard form
• 11Counting molecules in a mass of water

Worked examples (12)extended set

• 1Converting between two currencies
• 2Conversion with commission
• 3Best value across three different pack sizes
• 4Conversion via a third currency
• 5Profit and loss in a conversion context
• 6Reverse percentage with money
• 7Using a currency conversion graph
• 8Cheapest country for the same item
• 9Fixed fee or worse rate? Two bureaux on one graph
• 10Ordering lengths given in different units
• 11Comparing prices per litre in two currencies
• 12Converting area and volume units

Worked examples (13)extended set

• 1Applying the index laws
• 2Evaluating negative and fractional indices
• 3Solving an index equation
• 4Simplifying surds
• 5Rationalising a denominator
• 6Expanding brackets containing surds
• 7Exact hypotenuse and area of a surd triangle
• 8Rectangle with conjugate surd sides
• 9Paper sizes: the √2 ratio and fractional indices
• 10Index equation with three different bases
• 11Fractional and negative indices on a whole term
• 12Expanding surd brackets to find c and k
• 13Exact perimeter of a quadrilateral from two right-angled triangles

Worked examples (11)extended set

• 1Writing bounds for a rounded measurement
• 2Bounds of an area (multiplication)
• 3Bounds of a difference (subtraction)
• 4Bounds of a speed (division)
• 5Bounds with a nearest-whole-unit measurement
• 6Deciding a sensible answer from bounds
• 7Reading bounds from a number line
• 8Bounds for a trapezium-shaped lawn
• 9Bounds in a right-angled triangle
• 10Smallest possible depth of water in a pond
• 11Greatest possible density of a ball bearing

Worked examples (11)extended set

• 1Compound interest
• 2Compound interest against simple interest
• 3Depreciation
• 4Reverse problem — finding the original amount
• 5Finding the number of years
• 6Repeated halving (decay)
• 7Reading a decay curve: caffeine in the body
• 8Finding a growth model from two points on a curve
• 9A bouncing ball: heights, a threshold and total distance
• 10Finding a cooling model from two readings
• 11The interest rate that trebles an investment

Worked examples (10)extended set

• 1Speed, distance and time
• 2Average speed over two stages
• 3Converting between rate units
• 4Density of an alloy
• 5Rate of flow
• 6Rate in a costing context
• 7Pressure of a crate on different faces
• 8Filling a part-full cylindrical tank
• 9Reading tap and drain rates from a graph
• 10Time to drain a cylindrical tank, in minutes and seconds